Case Strength Check

The case strength check runs textbook strength formulas on your motor case. It covers the tube, plus how the forward closure (bulkhead) and the aft closure (nozzle) are held in it: snap rings in a groove, threads, or radial bolts. It is on the Case tab of the Motor & Nozzle section, it was added in 4.0.0.25, and it is free with every license.

This page covers every formula the check uses, every input, where each formula comes from, the limits on each one, and what is not checked at all. It ends with two examples worked in full. In the program, Show the working... on the Case tab shows the same thing for your own motor, with your own numbers in it.

These are hand calculations, not a safety assessment. They use textbook formulas with the inputs you typed. They don't know your actual material, your machining, your seals, your welds, or how the motor is built. Real cases fail in ways these formulas don't model. A number here is not evidence that a motor will hold. Testing is what tells you that. Nothing here is advice and nothing here is a guarantee.

That notice is printed on the Case tab, at the top of the working, and on the case page of the printed report. It cannot be turned off.

What it is, and what it is not

The check is arithmetic. It takes the dimensions and strengths you give it, applies the formulas below, and reports the pressure at which each formula says the part reaches its limit. It then compares the lowest of those pressures with your motor's MEOP (maximum expected operating pressure).

It never says a case is safe, never says it passes, and has no green tick. The result is worded as arithmetic about the design factors you entered:

Case check:      MEOP 1,000 psi is within the design factors entered. Calculation only, not a safety assessment (Case tab).

The other wording is MEOP ... is OUTSIDE the design factors entered. Both statements are about the numbers you typed. Neither one tells you whether the real case will hold.

The method reproduces Richard Nakka's CASING.XLS spreadsheet (v1.02/v1.03) for the tube. The snap-ring checks follow a workbook built on CASING.XLS, the one the author uses for his own cases. Threads and bolts use standard joint formulas, cited where they are used below. The first worked example reproduces that workbook's summary to the psi, and the second reproduces Nakka's own verification case.

Using the Case tab

Open the Motor & Nozzle section and click the Case tab, beside Motor. A motor starts with no case, and nothing is calculated until you press Define a case for this motor. Remove the case from this motor takes it out again.

The tab runs top to bottom:

1. Commercial case - a dropdown that fills in a commercial case's dimensions. Choose None - typed in by hand to enter your own. See below.
2. Tube - outside diameter, nominal wall thickness, wall tolerance, and whether to analyse at the minimum wall.
3. Material - picked from your materials library, with Edit materials... beside it.
4. Temperature - the design wall temperature, which derates the material.
5. Load - MEOP and the pressure-acting diameter.
6. Design factors - the five safety factors.
7. Forward closure (bulkhead) and Aft closure (nozzle) - one retention method each, with only that method's fields shown.
8. Results - calculation only - every check with its allowable pressure, then the notes and the not-checked list.

Show the working... opens the full calculation as text, which can be copied and printed.

Refused, not half-answered

If anything the check needs is missing, or the geometry cannot exist, the check does not run. The tab shows one message saying what is wrong, and gives no numbers at all. A partial answer would read as a result for the whole case. These all refuse the run:

• no MEOP (no simulation yet and none entered), no material, or no outside diameter or wall;
• a wall tolerance outside 0 up to (not including) 1, or any design factor of zero or less;
• a wall thicker than the tube's radius;
• a snap ring closure with no groove depth, or a groove at least as deep as the design wall;
• a thread with no major diameter, pitch or engaged length, or a thread larger than the tube's outside or smaller than its bore;
• radial bolts with no count, diameter, edge distance or bolt strength, a hole that breaks out of the tube end, holes wider than their spacing, or a ring with an edge distance but no bolts;
• a material whose yield-to-ultimate ratio (after derating) falls outside 0.5 to 1.0, the range the burst-factor curve was fitted over.

A closure method of none is a legitimate choice, and it is not refused. That closure is then not checked, and the tab says so.

Inputs

Tube

Field	Symbol	Notes
Outside diameter	Do	Measure it; don't trust the nominal size.
Wall thickness (nominal)	tnom
Wall tolerance	tol	A fraction. Drawn tube is commonly ±10%; check your tube's specification.
Analyse at the minimum wall		On by default. The check then uses t = tnom × (1 − tol), which is the correct choice for a part that has to hold pressure. Untick it to use the nominal wall.

Derived from these:

t_min = t_nom × (1 − tol)

t = t_min (or t_nom if the box is unticked) - the design wall every check below uses

D_i = D_o − 2t, r_o = D_o/2, r_i = D_i/2

A_wall = π(r_o² − r_i²)

D_o/t is reported too, because the thin-wall formula is only valid for a reasonably thin wall (Nakka uses D_o/t ≥ 20). The thick-wall formula has no such limit, and it is the one the result uses when it is lower.

If the bore is narrower than your widest grain, the tab warns you. The case as entered could not hold those grains.

Material

The material comes from your case materials library (Settings → Edit Case Materials..., or Edit materials... on the tab). Each material has:

• F_ty, tensile yield strength; F_tu, tensile ultimate strength; E, modulus; ν, Poisson's ratio;
• a specification, a source and an optional note;
• a temperature derating curve, or none.

BurnSim ships aluminium tube only. The figures are the aluminium rows of Nakka's CASING.XLS v1.03 Material Properties sheet, which cites MIL-HDBK-5G. They are typical values, not guaranteed minimums, and the working says so beside every one.

Material	Fty (ksi)	Ftu (ksi)	E (Msi)	ν	Derating curve
Aluminium 6061-T4 drawn tube	16	30	9.9	0.33	6061-T6
Aluminium 6061-T6 drawn tube (the default)	35	42	9.9	0.33	6061-T6
Aluminium 6061-T6511 extruded tube	35	38	9.9	0.33	6061-T6
Aluminium 2024-T3 drawn tube	45	66	10.5	0.33	2024-T3
Aluminium 2024-T42 drawn tube	38	62	10.5	0.33	2024-T3
Aluminium 7075-T6 drawn tube	66	77	10.4	0.33	none: enter derate factors by hand
Aluminium 7075-T73 drawn tube	56	66	10.4	0.33	none: enter derate factors by hand

Every number is editable, including each derating curve, and you can add your own materials. A built-in row you have edited is flagged in the working as no longer the published figures. Restore defaults brings the shipped rows back.

A motor keeps its own copy of its material, not a link to the library. Editing the library later never changes a saved motor's numbers, and a .bsx you share gives the same result on someone else's computer. The Case tab offers Use the library's current values when the library has moved on since the motor was saved.

Temperature

Design wall temperature is the hottest the wall gets while it is under pressure. Room temperature (80 °F) takes nothing off. Above that, F_ty and F_tu are multiplied by derate factors k_y and k_u, read off the material's curve with straight-line interpolation between the tabulated points. The curves are MIL-HDBK-5G's, for half an hour of exposure, from the Elevated Temperatures sheet of CASING.XLS:

Temperature (°F)	80	200	300	400	500
6061-T6 curve, ky (%)	100	92.2	83.3	69.5	44
6061-T6 curve, ku (%)	100	91	80	64	42
2024-T3 curve, ky (%)	100	94.2	87	77	50
2024-T3 curve, ku (%)	100	92	84	75	50

F_ty,eff = F_ty × k_y(T), F_tu,eff = F_tu × k_u(T)

A temperature beyond either end of the curve is clamped to that end, and the working says so. A material with no curve (the 7075 rows, or one of your own) takes the Yield derate factor and Ultimate derate factor you type in, and 1 means no reduction. You can also tick Enter the strength derate factors by hand to override any curve.

Load

MEOP, the maximum expected operating pressure. Left blank, it is the last simulation's peak chamber pressure. With a commercial license, if the experimental Ignition tab has a result with a higher peak, that peak is used instead and the source line says so. Type a value to override either one. The tab always says where the MEOP it used came from.

Pressure-acting diameter D_p is the diameter the pressure pushes on to load each closure. It defaults to the tube bore. Enter a larger one if your seal sits in a groove outside the bore.

A_p = π/4 × D_p²

F = MEOP × A_p, the end load on each closure

The whole end load on a closure passes through its retainer, which is simple equilibrium. Every closure check below is a check of how that one force F is carried.

Design factors

Factor	Default	Applies to
SD, on yield	1.5	The tube wall and the net sections at grooves, threads and bolt rows. Nakka recommends 1.5 to 2.0, and higher for a material of unknown strength.
SF, snap ring shear	2	The ring shearing through its own thickness.
SF, groove wall bearing	2	The groove edge yielding under the ring, and bolt holes bearing in the tube. Groove-edge rollover is the usual snap ring failure in aluminium. Don't go below 2.
SF, land shear-out, thread and bolt-hole tear-out	2	Shear through the case wall: the land ahead of a groove, a stripped thread, or a bolt tearing out to the tube end.
SF, radial bolt shear	2	The bolts themselves shearing.

Each allowable pressure is the failure pressure of its formula divided by its factor. The margin is reported as allowable / MEOP − 1, so 0.89 means the allowable is 89% above MEOP, and a negative margin means MEOP is above the allowable.

The tube

The plain tube wall, away from any closure, is checked two ways, and the lower result is the one that counts.

Thin wall: Barlow's formula on the outside diameter

P_y = 2 t F_ty,eff / D_o

PD = P_y / SD

This is Nakka's CASING.XLS method. Using the outside diameter rather than the mean or inside one makes it the conservative form of Barlow's formula. It comes from balancing the hoop force on a half-tube: the pressure across the diameter against the wall stress across two wall thicknesses.

Thick wall: Lamé, with the Tresca yield criterion

The exact elastic stresses in a thick-walled cylinder under internal pressure P (Lamé; Roark, Formulas for Stress and Strain, Table 32) are largest at the bore:

hoop σ_θ = P (r_o² + r_i²) / (r_o² − r_i²)

radial σ_r = −P

axial, closed ends, σ_z = P r_i² / (r_o² − r_i²)

Tresca says the material yields when the largest difference between two principal stresses reaches F_ty. At the bore that difference is hoop minus radial:

σ_θ − σ_r = P (r_o² + r_i²)/(r_o² − r_i²) + P = 2 P r_o² / (r_o² − r_i²)

Setting that equal to F_ty,eff and solving for P:

P_y = F_ty,eff (r_o² − r_i²) / (2 r_o²)

PD = P_y / SD

Von Mises is shown beside it for comparison but is not used, because it is less conservative. Write a = r_o²/(r_o² − r_i²) and b = r_i²/(r_o² − r_i²). Then σ_θ = P(a+b), σ_r = −P(a−b) and σ_z = Pb. The three differences are Pa, Pa and −2Pa, so

σ_vm = √(½[(Pa)² + (Pa)² + (2Pa)²]) = √3 · P a

P_y(von Mises) = F_ty,eff (r_o² − r_i²) / (√3 r_o²)

The working also prints P_y(Barlow)/P_y(Tresca). Above 1 means the thin-wall result is the less conservative of the two.

Burst

The burst pressure is the pressure at which the tube ruptures, as opposed to first yielding. It is reported for sizing a pressure relief device, which has to release well below it. It is not in the governing result.

b = F_ty,eff / F_tu,eff

B = 9.5833 b⁴ − 33.528 b³ + 44.929 b² − 28.479 b + 8.6475

PU = P_y(Barlow) × B

SU = PU / PD, the burst factor of safety

B is a curve fit of the burst factor in Nakka's CASING.XLS (Analysis sheet, T24:T33), after A. Blake, Practical Stress Analysis in Engineering Design. It was fitted for 0.5 ≤ b ≤ 1.0 only, which is why a material outside that range refuses the run.

The headline burst pressure is the lowest rupture pressure that applies: the plain tube's, or that of a groove, thread or bolt row that chamber pressure reaches (see below), whichever is lower.

Stress state, growth and cross-checks

These are printed for reference. They are not limits.

• Stress at MEOP: the Lamé hoop stress at the bore, the axial stress F/A_wall, the radial stress −MEOP, and their von Mises combination, √(½[(σ_h−σ_a)² + (σ_a−σ_r)² + (σ_r−σ_h)²]). It also gives the margin on yield, F_ty,eff/σ_vm − 1, with and without SD.
• Elastic radial growth of a closed-end thin cylinder (Nakka, after Roark): ΔD = P D_o² / (2 E t) × (1 − ν/2), at MEOP and at PD, plus the change in circumference πΔD. It is elastic, not permanent, and is useful for judging how much a liner or grain must give.

Snap ring closures

A ring sits in a groove cut into the tube. The closure bears on the ring, and the ring bears on the groove's edge. The fields:

Field	Symbol
Groove depth (radial depth below the tube surface)	d
Groove width (axial)	w
Groove root corner radius	r
Land length, groove edge to tube end (the material the ring would shear out of the tube end)	L
Ring thickness (axial)	tring
Ring radial engagement (how far the ring actually bears in the groove: the catalogue figure, not the groove depth)	h
Ring ultimate strength (from the ring catalogue; carbon spring steel SAE 1060-1090 is typically 170-200 ksi, 302 stainless about 125 ksi)	Ftu,ring
Ring shear as a fraction of ultimate (default 0.6)	f
Chamber pressure reaches the retainer (tick it if the seal is outboard of the groove)

Choose Snap ring, internal groove for a groove cut in the bore, or Snap ring, external groove for one cut in the outside of the tube.

Groove geometry

t_net = t − d, the wall left at the groove

internal groove: r_o' = D_o/2 and r_i' = D_i/2 + d; external groove: r_o' = D_o/2 − d and r_i' = D_i/2

A_net = π(r_o'² − r_i'²)

D_s, the ring's shear and bearing diameter: the bore D_i for an internal groove, the outside D_o for an external one

A_net is conservative, because it takes no credit for the full-thickness wall on either side of a narrow groove.

The checks

Each check turns a stress limit into the pressure that would produce it: the allowable force through the part, divided by A_p.

Check	Allowable pressure	What fails
Groove axial net section	P = Fty,eff Anet / Ap / SD	The end load pulls the reduced wall at the groove in tension. This applies whether or not pressure reaches the groove. The same relation as Half Cat Rocketry p.60, σ = Di² MEOP / (Do² − Dgroove²).
Snap ring shear	P = (f Ftu,ring / SFring) × π Ds tring / Ap	The ring shears through its own thickness. Compare it with the ring maker's catalogue rating (Smalley, Rotor Clip).
Groove wall bearing	P = (Fty,eff / SFbearing) × π Ds h / Ap	The groove edge yields and the ring rolls out. This is the usual snap ring failure in aluminium. It assumes uniform bearing over the engaged depth. Real contact concentrates at the groove edge, so treat SF = 2 as a floor.
Land shear-out	P = (0.577 Fty,eff / SFshear) × π Ds L / Ap	The ring tears the end off the tube, shearing out the land between the groove and the tube end. 0.577 Fty is the von Mises shear yield (1/√3).

Hoop at the groove, and seal bypass

Pressure inside the groove would load the thinned wall in hoop. The check uses the same two formulas as the tube, applied to the net section:

thin wall: P_y = 2 t_net F_ty,eff / D_o,eff

Lamé Tresca: P_y = F_ty,eff (r_o'² − r_i'²) / (2 r_o'²), the figure used

PD = P_y / SD

• If Chamber pressure reaches the retainer is ticked, this is a real limit. It counts toward the governing result, and a burst line is added for the groove.
• If not, it is listed separately under If the seal is bypassed (not counted above), and it is always printed. A nicked or extruded O-ring, or a liner burn-through, puts chamber pressure into a groove that was never meant to see it. A seal failure must not hide behind the setting. In the worked example below, that line is below MEOP.

Stress at the groove, and rules of thumb

For reference, the working gives the axial stress F/A_net, the hoop stress if the groove is pressurized, their von Mises combination, and the margins on yield.

It also gives a stress concentration estimate, K_t = 1 + 2√(d/r) (the Inglis notch formula). This is a deliberate upper bound: a real circumferential groove is usually K_t 2 to 3 (Peterson, Stress Concentration Factors). It is for fatigue reference only. A ductile alloy yields locally and redistributes the load, so it does not reduce static capacity.

Two rules of thumb from A. Sennott and C. Sharp, How to Design Pressure Vessels, Propellant Tanks, and Rocket Motor Casings (Half Cat Rocketry, p.60, after Nakka), are shown as cross-checks, not limits:

• Land length E_min = MEOP × D_s / (0.577 F_ty,eff), with 2 to 3 × E_min recommended for small casings. BurnSim uses 3. Compare it with the land shear-out formula: π D_s L τ = MEOP × π D_s²/4 gives L = MEOP D_s/(4τ). E_min leaves out the 4, so it carries roughly 4× margin.
• Groove depth d ≤ t/2.

The Notes block says where your land length and groove depth fall against both.

Threaded closures

The closure screws into the tube. The fields are Thread major diameter D, Thread pitch p (entered as threads per inch in imperial mode, or in mm in metric), Engaged thread length L_e, and optionally the Closure yield strength. That last one is used for stripping the closure's own thread. Left blank, it assumes the closure is the tube's material.

Thread geometry

For a 60° thread (UN and ISO metric share the same basic profile, ASME B1.1):

E = D − 0.649519 p, the basic pitch diameter

D₁ = D − 1.082532 p, the basic minor diameter of the internal thread

n = 1/p threads per inch; L_e/p threads engaged

t_net = (D_o − D)/2, the wall left under the thread, since the case's internal thread is cut out to D

The checks

Check	Allowable pressure	What fails
Case thread stripping	An = π n Le D [1/(2n) + 0.57735 (D − E)], P = (0.577 Fty,eff / SFshear) An / Ap	The case's internal thread shears off along the closure's thread crests. At basic dimensions An = 0.875 π D Le.
Closure thread stripping	As = π n Le D1 [1/(2n) + 0.57735 (E − D1)], P = (0.577 Fty,closure / SFshear) As / Ap	The closure's external thread shears off along the case's crests. At basic dimensions As = 0.75 π D1 Le.
Thread axial net section	Anet = π/4 (Do² − D²), P = Fty,eff Anet / Ap / SD	The end load pulls the wall left under the thread in tension.
Thread hoop at net section	Py = Fty,eff (ro² − (D/2)²) / (2 ro²), PD = Py/SD	A limit only if pressure reaches the thread, and then a burst line is added for it too. Otherwise it is the seal-bypass line, as for a groove.

The shear areas are the thread-stripping areas from Machinery's Handbook (Length of Engagement) and FED-STD-H28/2, evaluated at basic dimensions. Two stated limits are printed with every threaded result:

• A real 2A/2B fit has clearance that makes both shear areas smaller.
• The load is not shared evenly along the engagement: the first engaged threads carry more than their share.

Neither effect is in the numbers. Fine threads in thin walls are where this matters most.

Radial bolts

Bolts or screws pass radially through the tube wall into the closure. The fields are:

• Bolt diameter (all rings) d, the nominal size. The hole in the tube is taken as this size, with no clearance.
• Bolt thread pitch in the shear plane p. Leave it blank (0) if the plain shank is in the shear plane. This is a stated choice, not missing data.
• Bolt ultimate strength F_tu,bolt. It is never guessed: if it is blank, the run is refused.
• Bolt shear as a fraction of ultimate f, default 0.6 (about 0.6 for steel; Shigley, Machinery's Handbook).
• Up to four rings of bolts, each with its own Bolts in ring count N_k and Edge distance e_k from the hole centre to the tube end.

N is the total bolt count. Every bolt carries F/N, whichever ring it is in.

Check	Allowable pressure	What fails
Bolt shear	ds = d − 1.226869 p (or d for a plain shank), τallow = f Ftu,bolt / SFbolt, P = τallow N π/4 ds² / Ap	The bolts shear off at the joint, each in single shear. ds with a thread in the plane is the ASME B1.1 minor diameter of an external thread (rounded root). Bolt strength is not derated for temperature.
Bolt hole bearing in the tube	P = (Fty,eff / SFbearing) N d t / Ap	The bolts crush the edges of their holes. This is conservative: published bearing yield strengths (MMPDS) are well above Fty at e/d of 1.5 or more.
Bolt hole tear-out, per ring	A = Nk × 2 (ek − d/2) t, P = (0.577 Fty,eff / SFshear) A / (Ap Nk/N)	Each bolt shears a plug out of the wall toward the tube end, along two planes tangent to its hole. Each ring carries its Nk/N share of the load.
Bolt row axial net section, per ring	Anet = π/4 (Do² − Di²) − Nk d t, P = Fty,eff Anet / Ap / SD	The end load pulls the wall left between the holes in tension. This is taken at the full end load for every ring, which is conservative, because a ring really sheds part of the load to the next.
Bolt row hoop at the ligament, per ring	s = π(Do − t)/Nk, η = (s − d)/s, Py = Py(Lamé Tresca, plain wall) × η, PD = Py/SD	The plain-wall hoop figure is scaled by the ligament efficiency, the share of the hoop section left between neighbouring holes. It is a limit if pressure reaches the bolts, with a burst line added, and otherwise the seal-bypass line.

For reference, the working gives K_t = 3 at a round hole (Peterson) for fatigue, and e/d for each ring. A note is added below 2 (Bruhn). With one ring the checks keep their plain names. With several rings they are named per ring, and rings closer than 2 bolt diameters apart get a note.

Not checked for bolts: the closure side of the joint, meaning the bolts' bearing in the bulkhead or nozzle body and tear-out of the hole there. Make that part at least as strong as the tube. With several rings, an inner ring's plug passing out through an outer ring's holes is not modelled either.

The result

• Design pressure - the lowest allowable pressure among every check that limits pressure, with the check it came from. That check governs.
• Burst pressure - the lowest rupture pressure that applies (the plain tube, and any groove, thread or bolt row that chamber pressure reaches), for sizing a pressure relief device.
• MEOP, where it came from, and the margin of the governing check.
• The sentence about the design factors, as quoted at the top of this page.
• Every check, with its allowable pressure and margin. The seal-bypass lines and the burst lines are listed separately and do not count toward the design pressure.
• Notes - the rules of thumb, clamped temperatures, and anything else worth knowing about these numbers.
• Not checked here - the list below, beside the results rather than on another page. A check that doesn't exist reads as a check that passed if nobody is told it's missing.

When a case is defined, the Results tab adds three lines at the bottom (design pressure, burst pressure, and the MEOP sentence), and the motor report adds one or two pages after the motor page. Those pages carry the notice, the result, a table of every check, the inputs, the notes and the not-checked list. A refused run prints the refusal and nothing else. The formulas themselves stay in Show the working.

Every calculation runs in inches, psi and lbf, whatever units BurnSim is displaying. In metric mode the Case tab, the Results lines and the report show the same results in mm and kPa.

Commercial cases

The Commercial case dropdown fills in the dimensions of a commercial reloadable case. Each value comes with its source. The library ships:

• AeroTech RMS cases, 29, 38, 54, 75 and 98 mm (36 cases), with threaded closures at both ends. Outside diameters and lengths come from AeroTech's dimensional drawings.
• Loki Research snap ring cases (17).
• Kosdon snap ring cases (5).

All commercial cases are 6061-T6. Where a figure is not published and nobody has measured it, it is entered as 0 and the tab asks you for it rather than guessing. In 4.0.0.25 the AeroTech 98 mm bore and thread, and the Loki snap ring dimensions, are stand-ins until measured, and they say so in their source notes.

Settings → Edit Case Library... (or Edit cases... on the tab) lets you correct an entry or add your own cases. Your library is a file beside your settings. Built-in rows you have not edited refresh from each new BurnSim release, and an edited one is flagged as yours. As with materials, a motor keeps its own copy of the case values and their sources.

Case fit: the grains and the ullage

A case that knows how much propellant it takes (its liner, or the maker's maximum propellant length) is compared with your grain stack:

• If the grains stack longer than the case takes, the Case tab, the Results tab (Case fit: WARNING - ...) and the report all say by how much. Nothing is trimmed.
• If they are shorter, the spare length is split evenly into the motor's forward and aft ullage on the Motor tab, while Set the motor's ullage from this case is ticked. Choosing such a case ticks it. Untick it to enter the ullage yourself.

This is arithmetic on lengths only. The strength check doesn't use any of it.

What is not checked

This list is printed beside the results and in the report:

• Fatigue. K_t is shown at a groove or a hole for reference only. Repeated firings on a sharp groove need a real fatigue analysis.
• Buckling, from external pressure or from axial and bending flight loads.
• Welds and heat-affected zones. If the tube is welded, knock F_ty and F_tu down yourself.
• Snap ring dishing or rotation out of the groove, ring free diameter against groove fit, and the ring maker's own load rating. Take these from the catalogue (Smalley, Rotor Clip).
• Any hole, port, vent, set screw or thread in the tube other than the closures configured here.
• Thermal gradients through the wall, creep at temperature, and whether the liner or insulation keeps the wall at the design temperature.
• The bulkhead and nozzle as parts: plate bending, nozzle ejection, throat erosion.
• Seals: O-ring extrusion, and whether the seal really sits inboard of the retainer.
• Flight loads, impact, transport and handling.
• Non-metallic cases. The formulas assume an isotropic metal. Phenolic, fibreglass, composite and paper tubes are not isotropic, and these formulas do not describe them.

Worked example 1: snap ring closures

This is the reference case the check was built against. It reproduces the source workbook's summary to the whole psi.

Inputs. A 2.125 in OD × 0.100 in wall tube of 6061-T6 drawn tube (F_ty 35,000 psi, F_tu 42,000 psi, E 9.9 Msi, ν 0.33). Wall tolerance is 10%, analysed at the minimum wall, at 80 °F, with MEOP 1,000 psi. Default factors: SD 1.5, and 2 for ring shear, bearing and shear-out. Both closures use an internal groove 0.05 in deep, 0.06 in wide, with a 0.01 in root radius and a 0.25 in land. Each holds a 0.093 in ring engaged 0.08 in, F_tu,ring 180,000 psi, shear fraction 0.6. Neither groove is pressurized.

Derived values.

t = 0.100 × (1 − 0.10) = 0.090 in

D_i = 2.125 − 2 × 0.090 = 1.945 in; r_o = 1.0625, r_i = 0.9725

A_wall = π(1.0625² − 0.9725²) = 0.5754 in²; D_o/t = 23.6

k_y = k_u = 1 at 80 °F, so F_ty,eff = 35,000 and F_tu,eff = 42,000 psi

A_p = π/4 × 1.945² = 2.9712 in²; F = 1,000 × 2.9712 = 2,971.2 lbf

Tube.

Thin wall: P_y = 2 × 0.09 × 35,000 / 2.125 = 2,964.7; PD = 2,964.7 / 1.5 = 1,976.5 psi

Lamé Tresca: P_y = 35,000 × (1.0625² − 0.9725²) / (2 × 1.0625²) = 2,839.1; PD = 2,839.1 / 1.5 = 1,892.8 psi

Von Mises, for comparison: P_y = 35,000 × (1.0625² − 0.9725²) / (√3 × 1.0625²) = 3,278.4 psi

Barlow / Tresca = 2,964.7 / 2,839.1 = 1.044, so the thin-wall figure is the less conservative one here

Each groove (both are the same).

t_net = 0.09 − 0.05 = 0.04 in; r_o' = 1.0625, r_i' = 0.9725 + 0.05 = 1.0225

A_net = π(1.0625² − 1.0225²) = 0.26201 in²; D_s = D_i = 1.945 in

Axial net section: 35,000 × 0.26201 / 2.9712 = 3,086.4; / 1.5 = 2,057.6 psi

Ring shear: A = π × 1.945 × 0.093 = 0.56827 in²; τ_allow = 0.6 × 180,000 / 2 = 54,000; P = 54,000 × 0.56827 / 2.9712 = 10,328 psi

Groove wall bearing: A = π × 1.945 × 0.08 = 0.48883 in²; σ_allow = 35,000 / 2 = 17,500; P = 17,500 × 0.48883 / 2.9712 = 2,879.2 psi

Land shear-out: A = π × 1.945 × 0.25 = 1.5276 in²; τ_allow = 0.577 × 35,000 / 2 = 10,097.5; P = 10,097.5 × 1.5276 / 2.9712 = 5,191.5 psi

Seal bypass (hoop at the net section, Lamé Tresca): 35,000 × (1.0625² − 1.0225²) / (2 × 1.0625²) = 1,292.8; / 1.5 = 861.9 psi, margin −0.14. The thin-wall figure is 878.4 psi.

Burst. b = 35,000 / 42,000 = 0.8333; B = 1.3345; PU = 2,964.7 × 1.3345 = 3,956.4 psi; SU = 3,956.4 / 1,976.5 = 2.00.

Reference figures at MEOP. Hoop at the bore 11,328 psi, axial 5,164 psi, radial −1,000 psi, von Mises 10,676 psi, so the margin on yield is 2.28 with no factor and 1.19 with SD. Radial growth is 0.00212 in at MEOP and 0.00418 in at PD. At each groove: axial stress 11,340 psi and K_t = 1 + 2√(0.05/0.01) = 5.47. E_min = 1,000 × 1.945 / (0.577 × 35,000) = 0.0963 in, and 3 × E_min = 0.289 in.

Result. Design pressure 1,892.8 psi, governed by the tube wall (Lamé, Tresca), margin 0.89 against 1,000 psi. Burst pressure 3,956.4 psi. MEOP is below the design pressure, so by these formulas it is within the design factors entered. That is a statement about arithmetic, not about whether the case will hold.

The notes then say two things. First, each 0.25 in land lies between E_min and 3 × E_min. Second, each groove is 56% of the design wall, beyond the d ≤ t/2 rule of thumb's 0.045 in. And the seal-bypass line, 862 psi, is below MEOP. If either O-ring were bypassed, these formulas would put the thinned wall at the groove beyond its design factor at the motor's own pressure. That is exactly what the seal-bypass line exists to show.

The same tube with other closures

With the same tube and load, and the forward closure changed:

• A 2.000-12 thread engaged 0.5 in (6 threads, closure of the tube's material): case thread stripping 9,342 psi, closure thread stripping 7,646 psi, thread net section 3,180 psi, seal bypass 1,332 psi. The tube still governs at 1,892.8 psi.
• Four 1/4-20 bolts, threads in the shear plane, 170 ksi, 0.5 in from the end: bolt shear 1,919 psi, hole bearing 530 psi, tear-out 918 psi, net section 3,812 psi. Hole bearing governs, and MEOP is outside the design factors entered. Four bolts of that size are too few for this load, by these formulas.

Worked example 2: Nakka's verification case

This is CASING.XLS's own check case: D_o 4.375 in, wall 0.180 in at nominal (no tolerance), F_ty 40 ksi, F_tu 45 ksi, SD 1.5. Nakka gives no E; 10.0 Msi and ν 0.33 reproduce his growth figure.

P_y = 2 × 0.180 × 40,000 / 4.375 = 3,291.4; PD = 3,291.4 / 1.5 = 2,194 psi

b = 40/45 = 0.889, B = 1.267; PU = 3,291.4 × 1.267 = 4,171 psi; SU = 4,171 / 2,194 = 1.90

ΔD at PD = 2,194 × 4.375² / (2 × 10,000,000 × 0.180) × (1 − 0.33/2) = 0.00974 in

All four figures match CASING.XLS. Both worked examples run as automated tests on every BurnSim build, so a change to these formulas that moved any of these numbers would fail the build.

See also

• Accuracy and Validation - what BurnSim's other results are checked against
• BurnSim Fields - the Motor tab beside this one
• Results Tab and Motor Report - where the result appears outside the Case tab
• Richard Nakka's CASING spreadsheet - the method and spreadsheet the tube checks reproduce